I will provide here a short summary of my previous posting, and will add at

the end an even simpler geometrical interpretation of the definition of the

moment arm "of a force" F about a rotation axis w (more properly called the

moment arm about w "of the rejection of F from w").

But first, let me explain to everybody (as I already did in private to a few

subscribers) that, although both Ton and I provided a formula to compute

that moment arm, that formula was not meant to be used in practice in

biomechanics for computing the moment arm about a joint axis such as the

flexion-extension axis of the hip. It was just a mathematical definition. In

practice, typically we cannot compute F (muscle force), unless we first

measure or estimate the moment arm. So, it is in most cases impossible to

compute the moment arm from the values of F and M (or from F-Fw and Mw). For

instance, I used this orthogonal cross division to define the moment arm of

F about a point O:

- M /o F = (F × M) / (F * F) (de Leva, 2008)

but I did not invent the cross divisions to define moment arms in 3-D! The

practical applications of cross divisions do not include the computation of

the moment arm of a force (see my next message). Here is a brief summary of

my definitions of moment arm about any rotation axis w:

MATHEMATICAL DEFINITION (MOMENT ARM ABOUT A POINT)

- (Vector) Moment arm of F about O = M /o F

- (Scalar) moment arm of F about O = |M /o F|

where /o is the symbol for the orthogonal cross division. The traditional

method is equivalent, but has two drawbacks. It is computationally less

efficient, and can only be used to define the scalar moment arm:

- (Scalar) moment arm of F about O = |M| / |F|

MATHEMATICAL DEFINITION (MOMENT ARM ABOUT ANY ROTATION AXIS)

- (Vector) Moment arm of F–Fw about w = Mw /o (F–Fw)

- (Scalar) moment arm of F–Fw about w = |Mw /o (F–Fw)|

where

- w is any rotation axis w

- Mw is the projection of M on w

- Fw is the projection of F on w

- F–Fw is the rejection of F from w.

- /o is the symbol for the orthogonal cross division

Notice that w does not need to coincide with an axis of your coordinate

system. The traditional method is equivalent, but has two drawbacks. It is

computationally less efficient, and can only be used to define the scalar

moment arm:

- (Scalar) moment arm of F–Fw about w = |Mw| / |F–Fw|

GEOMETRICAL INTERPRETATION (MOMENT ARM ABOUT A POINT)

In general, the moment arm of a force F about a point O in 3-D is actually a

vector (it undeniably has a direction). If the moment of F is

- M = d × F

then the moment arm of F about O is just the component of d orthogonal to F

(d_orth). This is also called the orthogonal anticrosproduct of d × F (any

cross product has its unique orthogonal anticrossproduct). This moment arm

has a property: by the definition of the cross product,

- d_orth × F = d × F = M.

In most cases you need to know just the magnitude of the moment arm

(|d_orth|), which is often referred to simply as "the moment arm", rather

than its magnitude. That's a widely used terminological convention, so you

can use it, but don't forget that d_orth is a vector.

GEOMETRICAL INTERPRETATION (MOMENT ARM ABOUT ANY ROTATION AXIS)

Similarly, if the moment of F-Fw about any rotation axis w is

- Mw = (d-dw) × (F-Fw)

So, the moment arm of F-Fw about w is just the component of d-dw orthogonal

(i.e. perpendicular) to F. This is also called the orthogonal

anticrossproduct of the above-mentioned cross product, (d-dw) × (F-Fw).

EVEN SIMPLER GEOMETRICAL INTERPRETATION

Interestingly, we can also define the moment arm of F-Fw about w as the

component of d orthogonal to F-Fw.

Unfortunately, we cannot further simplify the definition. Indeed, the

component of d orthogonal to F (d_orth) is something else. Namely, it is the

moment arm of F about O (see above), typically longer than the moment arm of

F-Fw about w.

REFERENCE

de Leva P., 2008. Anticrossproducts and cross divisions. Journal of

Biomechanics, 8, 1790-1800

(http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030)

With kind regards,

Paolo de Leva

Laboratory of Locomotor Apparatus Bioengineering

Department of Human Movement and Sport Sciences

University of Rome, Foro Italico

Piazza Lauro de Bosis, 6

00135 Rome - Italy

http://www.lablab.eu

the end an even simpler geometrical interpretation of the definition of the

moment arm "of a force" F about a rotation axis w (more properly called the

moment arm about w "of the rejection of F from w").

But first, let me explain to everybody (as I already did in private to a few

subscribers) that, although both Ton and I provided a formula to compute

that moment arm, that formula was not meant to be used in practice in

biomechanics for computing the moment arm about a joint axis such as the

flexion-extension axis of the hip. It was just a mathematical definition. In

practice, typically we cannot compute F (muscle force), unless we first

measure or estimate the moment arm. So, it is in most cases impossible to

compute the moment arm from the values of F and M (or from F-Fw and Mw). For

instance, I used this orthogonal cross division to define the moment arm of

F about a point O:

- M /o F = (F × M) / (F * F) (de Leva, 2008)

but I did not invent the cross divisions to define moment arms in 3-D! The

practical applications of cross divisions do not include the computation of

the moment arm of a force (see my next message). Here is a brief summary of

my definitions of moment arm about any rotation axis w:

MATHEMATICAL DEFINITION (MOMENT ARM ABOUT A POINT)

- (Vector) Moment arm of F about O = M /o F

- (Scalar) moment arm of F about O = |M /o F|

where /o is the symbol for the orthogonal cross division. The traditional

method is equivalent, but has two drawbacks. It is computationally less

efficient, and can only be used to define the scalar moment arm:

- (Scalar) moment arm of F about O = |M| / |F|

MATHEMATICAL DEFINITION (MOMENT ARM ABOUT ANY ROTATION AXIS)

- (Vector) Moment arm of F–Fw about w = Mw /o (F–Fw)

- (Scalar) moment arm of F–Fw about w = |Mw /o (F–Fw)|

where

- w is any rotation axis w

- Mw is the projection of M on w

- Fw is the projection of F on w

- F–Fw is the rejection of F from w.

- /o is the symbol for the orthogonal cross division

Notice that w does not need to coincide with an axis of your coordinate

system. The traditional method is equivalent, but has two drawbacks. It is

computationally less efficient, and can only be used to define the scalar

moment arm:

- (Scalar) moment arm of F–Fw about w = |Mw| / |F–Fw|

GEOMETRICAL INTERPRETATION (MOMENT ARM ABOUT A POINT)

In general, the moment arm of a force F about a point O in 3-D is actually a

vector (it undeniably has a direction). If the moment of F is

- M = d × F

then the moment arm of F about O is just the component of d orthogonal to F

(d_orth). This is also called the orthogonal anticrosproduct of d × F (any

cross product has its unique orthogonal anticrossproduct). This moment arm

has a property: by the definition of the cross product,

- d_orth × F = d × F = M.

In most cases you need to know just the magnitude of the moment arm

(|d_orth|), which is often referred to simply as "the moment arm", rather

than its magnitude. That's a widely used terminological convention, so you

can use it, but don't forget that d_orth is a vector.

GEOMETRICAL INTERPRETATION (MOMENT ARM ABOUT ANY ROTATION AXIS)

Similarly, if the moment of F-Fw about any rotation axis w is

- Mw = (d-dw) × (F-Fw)

So, the moment arm of F-Fw about w is just the component of d-dw orthogonal

(i.e. perpendicular) to F. This is also called the orthogonal

anticrossproduct of the above-mentioned cross product, (d-dw) × (F-Fw).

EVEN SIMPLER GEOMETRICAL INTERPRETATION

Interestingly, we can also define the moment arm of F-Fw about w as the

component of d orthogonal to F-Fw.

Unfortunately, we cannot further simplify the definition. Indeed, the

component of d orthogonal to F (d_orth) is something else. Namely, it is the

moment arm of F about O (see above), typically longer than the moment arm of

F-Fw about w.

REFERENCE

de Leva P., 2008. Anticrossproducts and cross divisions. Journal of

Biomechanics, 8, 1790-1800

(http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030)

With kind regards,

Paolo de Leva

Laboratory of Locomotor Apparatus Bioengineering

Department of Human Movement and Sport Sciences

University of Rome, Foro Italico

Piazza Lauro de Bosis, 6

00135 Rome - Italy

http://www.lablab.eu